Suggested Response - Q5

** Use Fourier’s law for steady state heat conduction to determine the rate of heat exchange from the ambient air into a stainless steel tank filled with beer at 1°C. You may assume the temperature of the air on the outside of the tank is 20°C, the thickness of the tank wall is 25.4 mm and the heat transfer surface area is 7.0 m2. The thermal conductivity of stainless steel is 17 Wm-1 K-1. Describe why it is important to know this value in a brewery.

Response:

First let’s recall Fourier’s law for steady state heat conduction:

q space equals space k A space fraction numerator increment T over denominator increment x end fraction

We are calculating the rate of heat gained from the air per heat transfer area. All values are given in the problem statement; we just have to determine if they are in the correct units and which values go where.

k space equals space t h e r m a l space c o n d u c t i v i t y space equals space 17 space W m to the power of negative 1 end exponent space K to the power of negative 1 end exponent

increment T space equals space T subscript 1 space minus space T subscript 2 space space space i s space t h e space t e m p e r a t u r e space d r o p space a c r o s s space t h e space l a y e r

increment T space equals space T subscript 1 space minus space T subscript 2 space equals space 20 degree C space minus space 1 degree C space equals space 19 degree C

increment x space equals space t h i c k n e s s space o f space t h e space l a y e r space equals space 25.4 space m m space equals space 0.0254 space m

Now we can enter the values and solve.

q space equals space 17 space cross times space 7 space cross times space fraction numerator 19 over denominator 0.0254 space end fraction space equals space 89 comma 016 space W space equals space 89 space k W

This value is the rate of heat gain from the air into the beer. It is useful in determining how much heat needs to be removed from the fermenter in order to keep the beer at 1°C.

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